November 13, 2012

Dimensional Analysis

Units (also called “dimensions”) are funny things, aren’t they? They actually change the MEANING of the numbers that they are attached to. For example, 3 cups isn’t the same as 3 gallons, even though both are “3 of something”. It’s something most people just take for granted. But both units are measures of the same physical concept – in this case, both are measures of volume. So there must be a way to convert from one (“cups”) to the other (“gallons”) and vice-versa.

When converting units, there is one overriding principle that you must adhere to – you are trying to change the UNITS, not the PHYSICAL QUANTITY of whatever is being measured. Most math students know that you use “conversion factors” to switch units, but what’s not obvious about these conversion factors is that they are all “equivalent to 1”. For example, I assert that 12 inches = 1 foot. Ok, “big deal” you say – everyone knows that. But it’s an interesting statement – I’m saying that “12 of something” is equivalent to “1 of something else”. Well, if we put those into a fraction form, then it looks weird but still holds true:

Therefore, I should be able to multiply ANY number out there by this ratio (12 in / 1 foot) and I’m not changing the answer, because all I’m doing is multiplying by 1, and everyone knows that multiplying by 1 doesn’t change the number. If that is true, then I should be able to write:

So what just happened? I started with a ratio, “3 feet over 1” and multiplied by a factor “equivalent to 1” and got a different answer. But I just said that multiplying by 1 doesn’t change the answer! So what has changed? Certainly the numerical value changed – the “3” is now “36” – and the units changed – “feet” is now “inches”. But 3 feet is equal to 36 inches, so we have successfully converted the original value of 3 feet to inches. In a sense, the numerical value that changed (3 to 36) exactly cancels out the units that changed (feet to inches). Notice how the units themselves actually seem to “cancel out” in the multiplication – “feet” (or “foot” in this case) in the denominator of the conversion factor cancel out with “feet” in the original value of “3 feet”. You can actually think of the units as “multiplication”, so that “3 feet” is really “3” times “feet”.

This is what “dimensional analysis” is all about – making “equivalent” changes to the numerical value and units at the same time, so as to keep the same amount of “stuff” that you started with. We do this by multiplying (or dividing) by conversion factors that are “equivalent to 1”. Along the way, the units that we are trying to get rid of are replaced by the units that we are trying to get to, and the numerical value changes in just the right way to keep the physical measurement the same.

The only hard part of dimensional analysis, if there is one, is knowing the correct “equivalent to 1” conversion factors for all of the different units out there. These conversion factors are often found in textbooks or online. Google is exceptionally good at helping to answer these questions – just search for something like “feet per mile” and Google will pop up not just the correct conversion, but the answer is often a mini-widget that you can use to convert many other units at the same time.

Here’s a more complicated example: How fast is 55 miles per hour in “inches per second”? There are a few different conversion factors we need here. First, we know that 1 mile = 5280 feet, 1 foot has 12 inches, 1 hour has 60 minutes, and 1 minute has 60 seconds. Thus, we could answer this question like so:

Note that there are a FEW cases in which you may multiply by something that is not “equivalent to 1”, but it’s usually due to how the problem is stated. For example: “How many tablespoons of sugar would you need to bake 5 times as many cookies as called for in a recipe that calls for 3 cups of sugar?” In this case, because the question is asking you to calculate “5 times as many” you are actually changing the amount of stuff you need. Thus, while the recipe calls for 3 cups of sugar, to make 5 times as many as expected in the recipe, you would need 3 * 5 = 15 cups. That can still be represented in dimensional analysis, but your factor will not be “equivalent to 1”:

There are a few other benefits of using the dimensional analysis technique:

Sometimes students have difficulty deciding when to “multiply by” or “divide by” a conversion factor. By making sure that the units properly cancel each other as you go, it is obvious when to multiply and when to divide.

By making sure that the units cancel out, in most cases, you can actually convince yourself that you have done the problem correctly. (Note: there are a few cases in which this doesn’t always work, but they are comparatively rare.)

Here’s an example of both. Using the “3 feet” example, let’s pretend we didn’t know whether to multiply or divide, and let’s pretend we guess incorrectly. We would end up with this:

In this case, the “feet” in are both in the numerator, so they cannot cancel, and we are also stuck with “inches” in the denominator. We know this is wrong because the units didn’t cancel correctly to give us our desired unit of “inches”. Therefore, we review our conversion factor and realize that “feet” must cancel “feet” because that’s what we are trying to get rid of, and “inches” must be left in the numerator because that’s what we are trying to get to. Therefore, the conversion factor “1 foot over 12 inches” is wrong, even though it is still “equivalent to 1”.